Regular Covering Spaces

Algebraic Topology · Axiom Academy

LESSON Regular Covering Spaces Normal subgroups and deck transformations 1. Definition of Regular Covers A covering p: E → B is regular if the image p₊(π₁(E, e₀)) is a normal subgroup of π₁(B, b₀). Equivalently, a covering is regular if for any two points e₁, e₂ ∈ p⁻¹(b₀), there exists a deck transformation (covering automorphism) taking e₁ to e₂. A deck transformation is a homeomorphism φ: E → E such that p ∘ φ = p. The set of all deck transformations forms a group Aut(E/B) called the group of covering transformations . For regular coverings, this group acts transitively on each fiber p⁻¹(b), meaning the covering "looks the same" from any point in a fiber. 3. The Fundamental Isomorphism For a regular covering, the automorphism group is isomorphic to the quotient of the fundamental group by the image of π₁(E): This shows that the symmetry group of the covering is determined entirely by group theory: it's the quotient by a normal subgroup. Let's examine some classic regular coverings: ℝ → S¹: Universal covers are always regular. Here Aut(ℝ/S¹) ≅ ℤ, generated by the translation t ↦ t + 1. z ↦ z²: S¹ → S¹: Regular with Aut ≅ ℤ/2ℤ, the deck transformation is z ↦ -z. z ↦ zⁿ: S¹ → S¹: Regular with Aut ≅ ℤ/nℤ, deck transformations are rotations by 2πk/n. ℝ² → T²: Universal cover of torus, Aut ≅ ℤ × ℤ generated by unit translations. Not all coverings are regular! For example, the inclusion of a wedge of two circles into a figure-eight with an extra loop is not regular.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.